AI 中文总结
该研究提出符号校正贝塞尔去偏（SCBD）伪边际MCMC方法，用于广义贝叶斯推理，结合普通MC与RQMC实现，可有效消除高斯指数膨胀，预算需求低，兼具方差减少与易编码优势。
AI 中文摘要
广义贝叶斯推理采用形式为$\text{exp}\{-\beta_n\ell_n(\theta)\}$的权重，即使损失仅被估计。对无偏损失估计取幂会改变目标，当$\beta_n\asymp n$时，方差阶为$M^{-1}$的普通蒙特卡洛损失估计需要每提议预算$M$为$n^2$阶，以保持主导对数权重方差有界。我们引入符号校正贝塞尔去偏（SCBD），一种基于损失的独立块估计的符号伪边际方法，并研究其普通MC和独立随机拟蒙特卡洛（RQMC）实现。在独立同分布高斯块模型下，由块样本方差构造的贝塞尔因子尽管方差未知，仍能精确消除高斯指数膨胀。对于一般非高斯有限块，该方法的目标后验与预期后验相差一个依赖参数的乘法因子。在正则条件下，未校正和校正的普通MC目标的总变差误差阶分别为$\beta_n^2/M_n$和$\beta_n^3/M_n^2$。若RQMC块估计量的方差为$\u004f\{B^{-\alpha}(\log B)^{d-1}\}$，则相应误差阶为$\text{delta}^{\text{RQ}}_{n,M_n}$和$(\delta^{\text{RQ}}_{n,M_n})^{3/2}$，其中$\text{delta}^{\text{RQ}}_{n,M_n}=\beta_n^2M_n^{-\alpha}{\log(2+M_n)}^{d-1}$。当$\beta_n\asymp n$时，相同方差率给出的充足预算阶为$n^{2/\alpha}$（对数因子除外），以保持主导对数权重方差有界。数值示例表明，良好的RQMC表示可继承该预算缩放，且方差减少与贝塞尔去偏互补。与现有精确校正相比，贝塞尔去偏本质上是“免费”的，具有通用性、易编码性且有理论支撑。
英文摘要
Generalised Bayesian inference uses weights of the form $\exp\{-β_n\ell_n(θ)\}$ even when the loss is only estimated. Exponentiating an unbiased loss estimate changes the target, and when $β_n\asymp n$ an ordinary Monte Carlo loss estimate with variance of order $M^{-1}$ requires a per-proposal budget $M$ of order $n^2$ to keep the leading log-weight variance bounded. We introduce Sign-Corrected Bessel Debiasing (SCBD), a signed pseudo-marginal method based on independent block estimates of the loss, and study its ordinary-MC and independently randomised quasi-Monte Carlo (RQMC) implementations. Under an i.i.d. Gaussian block model, a Bessel factor constructed from the block sample variance exactly removes the Gaussian exponential inflation despite the variance being unknown. For general non-Gaussian finite blocks, the method targets a posterior differing from the intended posterior by a parameter-dependent multiplicative factor. Under regularity conditions, the uncorrected and corrected ordinary-MC targets have total-variation errors of orders $β_n^2/M_n$ and $β_n^3/M_n^2$. If an RQMC block estimator has variance $\mathcal O\{B^{-α}(\log B)^{d-1}\}$, the corresponding errors are of orders $δ^{\mathrm{RQ}}_{n,M_n}$ and $(δ^{\mathrm{RQ}}_{n,M_n})^{3/2}$, where $δ^{\mathrm{RQ}}_{n,M_n}=β_n^2M_n^{-α}{\log(2+M_n)}^{d-1}$. The same variance rate gives a sufficient budget of order $n^{2/α}$, up to logarithmic factors, for bounded leading log-weight variance when $β_n\asymp n$. The numerical examples show that favourable RQMC representations can inherit this budget scaling and that variance reduction and Bessel correction are complementary. Compared to existing exact corrections, Bessel debiasing is essentially "for free". It is generic, easy to code and supported by theory.
Comments55 pages, 11 figures
